10 month old who isn't moving on own and is hypermobile

Sounds like a serious developmental problem. By 10 months baby should at least be crawling (or mobile in some way). I personally would casually point out things other babies in your group are doing. Or I would say "When my baby was 10 months he..." or "My neice didn't crawl till she was 9 months old..." It may be ackward, but you need to bring it up. Let her know you don't mean to offend or criticize, but you are worried about the baby. She may get angry at first, but will get over it. I'm a bit worried that she hasn't noticed this yet (especially considering her career). Perhaps she is already discussing with her doctor, but is embarresed to bring it up? Or she may be "blind" to her child being behind. Like they say "love is blind", but sometimes mommies can't admit to themselves something serious is wrong with their child. She may be too scared to find out, but you may give her the push she needs. If she does nothing about it you may want to contact child services to investigate.
You may also want to print out a "what to expect" at this age, show her, and tell her you were researching for yourself to help you care for the baby and thought she would be interested...

He can sit, but only if you sit him up. He doesn't do it on his own.

Yeah that is a touchy subject because even at the thought of anything being wrong parents usually FREAK OUT! I have a 9-month-old (first time mom), but I would say you should actually be more direct about it. I don't know when other kids do what as I don't have much experience with kids outside of my own (and what mma_mom helps me out with :)

As someone who takes her baby to child care 3 days a week I would want my daughters sitter to tell me if anything was out of the ordinary. I would just do it in a gentle way that wouldn't alarm her. Maybe say something like, I noticed Baby Jane isn't sitting by herself yet when she's here, does she do that for you at home? Maybe ask her if the pediatrician has said anything as she should have had a 9-month-old well baby check up.

Maybe just planting the idea will have her research on her own and maybe you can recommend giving a call to the pediatrician "just to be safe", while being comforting at the same time. Her and the pediatrician can deal with issues that arise. Good luck and you sound like a caring sitter - let us know what happens!